Problem 15 - Calculus for Biology and Medicine

Question

In Problems 15-20, assume that a quantitative character is normally distributed with mean \(\mu\) and standard deviation \(\sigma .\) Determine what fraction of the population falls into the given interval. \([\mu, \infty)\)

Step-by-Step Solution

Answer

The fraction of the population within the interval \([\mu, \infty)\) is 0.5 or 50%.

Step 1: Understanding the Normal Distribution

The normal distribution is a probability distribution with a symmetric, bell-shaped curve. It is defined by its mean (\(\mu\)) and standard deviation (\(\sigma\)). Due to its properties, we can easily calculate probabilities for different intervals.

Step 2: Identify the Interval

The problem asks to find the fraction of the population within the interval \([\mu, \infty)\). This means we want to find the probability that a randomly selected individual from the population will have a value greater than or equal to the mean \(\mu\).

Step 3: Standardize the Interval

To solve this, we need to use the standard normal distribution (a normal distribution with mean 0 and standard deviation 1). We convert the interval \([\mu, \infty)\) using the z-score formula: \(z = \frac{x - \mu}{\sigma}\). Here, \(x = \mu\), so \(z = \frac{\mu - \mu}{\sigma} = 0\).

Step 4: Determine Probability Using Z-Score

From the standard normal distribution, the probability that \(Z\geq 0\) (where \(Z\) is the standardized value) is the area to the right of \(Z=0\). This equates to 0.5 because the normal distribution is symmetric relative to the mean.

Key Concepts

Standard Deviation

Standard deviation () is a crucial concept in statistics that measures the amount of variation or dispersion in a set of data values. Essentially, it indicates how much the individual data points differ from the mean of the dataset.

A small standard deviation means data points tend to be close to the mean, indicating low variability. Conversely, a large standard deviation indicates that data points are spread out over a wider range, showing higher variability.

• Used in predicting data variability.
• Essential in calculating z-scores for standardization.
• Helps in understanding the spread of probability distributions.

Standard deviation plays a key role in the normal distribution, defining the bell shape and helping calculate the probability of specific intervals.

Z-score

The z-score is a statistical measurement that describes a value's position in relation to the mean of a group of values, measured in terms of standard deviations. The z-score indicates how many standard deviations an individual data point is from the mean.

• The z-score formula is: \( z = \frac{x - \mu}{\sigma} \)
• A z-score of 0 implies the value is at the mean.
• A positive z-score indicates a value above the mean, while a negative z-score is below.

You can transform any normal variable to a standard normal variable using its z-score, which is very handy in determining probabilities in a standard normal distribution. In our case, we calculated a z-score of 0 because the interval starts exactly at the mean (). This shows why understanding z-scores is essential to interpreting the probability of data within specific intervals in a normal distribution.

Probability Distribution

A probability distribution is a statistical function that describes the likelihood of obtaining possible outcomes of a random variable. A normal distribution is a specific type of probability distribution where data is symmetrically distributed, forming the shape of a bell curve.

The normal distribution

• Is determined by its mean and standard deviation.
• Has around 68% of the data within one standard deviation of the mean.
• Approximately 95% of the data falls within two standard deviations of the mean.

Understanding a probability distribution helps us determine the likelihood of a value within a given interval.
For example, probabilities can be easily calculated using the z-score under the normal distribution, like finding the chance of a value being greater than the mean or within a particular range.

Mean

The mean () of a data set is its average value and is a central measure in statistics that represents the typical value of the data. It is calculated by adding up all the values and dividing by the number of values.

• Provides a central point for the data.
• Essential in normal distributions where it defines the center of the bell curve.
• Used in computing standard deviation and z-scores to understand the data spread.

The mean is particularly significant for the normal distribution because this distribution is symmetric about the mean. As in the exercise, if we're investigating the probability of values within certain intervals, starting from the mean (), understanding the properties of the mean can serve as a baseline for further statistical analysis.